Abstract Algebra/Group Theory/Subgroup/Intersection of Subgroups is a Subgroup

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Theorem[edit]

Let H1, H2, ... Hn be subgroups of Group G with operation

with is a subgroup of Group G

Proof[edit]

[edit]

1. H1 is subgroup of G
2. H2 is subgroup of G
3. 1. and 2.


with is a Group [edit]

Closure[edit]

4. Choose
5. closure of H1
6. closure of H2
7. 5. and 6.


Associativity[edit]

8. is associative on G. Group G's operation is
9. 3.
10. is associative on 8. and 9.


Identity[edit]

11. and Subgroup H1 and H2 inherit identity from G
12. eG is identity of G,
13. and 9.
14. has identity eG definition of identity

Inverse[edit]

15. Choose
16. , , and
17. gH1-1 in H1, and gH2-1 in H2. G, H1, and H2 are groups
18.
19. G and H1 shares identity e
20. gH1-1 is inverse of g in G 19. and definition of inverse
21. Let gG-1 be inverse of g has in G
22. gG-1 = gH1-1 inverse is unique
22. gG-1 = gH2-1 similar to 21.
23.
24. g has inverse g-1 in